The Hole Probability for Gaussian Entire Functions

Abstract

We study the hole probability of Gaussian random entire functions. More specifically, we work with entire functions in Taylor series form with i.i.d complex Gaussian coefficients. A hole is the event where the function has no zeros in a disc of radius r. We find exact asymptotics for the rate of decay of the hole probability for large values of r, outside a small exceptional set (which is deterministic).

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